Among separable metrizable spaces:

Cantor set is the unique compact zero-dimensional space without isolated points.

$\mathbb Q$ is the unique countable space without isolated points

$\mathbb R \setminus \mathbb Q$ is the unique zero-dimensional, $G_\delta$-space with no compact neighborhood.

$\mathbb Q ^\omega$ is the unique zero- dimensional, first category $F_{\sigma\delta}$-space with the property that no nonempty clopen subset is a $G_{\delta\sigma}$-space.

**Question.** Is there a simple classification of zero-dimensional $G_{\delta\sigma}$-spaces which have no compact neighborhoods?
The simplest examples would be $\mathbb Q$, $\mathbb R\setminus \mathbb Q$, and $\mathbb Q\times (\mathbb R\setminus \mathbb Q)$. Are there many others?

Is there a nice characterization of $\mathbb Q\times (\mathbb R\setminus \mathbb Q)$?